The Microwave Background - TBC

Cosmology 2002/2003 1

The Microwave Background - TBC

Prof. Guido ChincariniThis part of the lectures introduce the MWB and ends stating three

problems which could be developed in many details and of great interest:

1) The Relation between the motion of the solar system and the distribution of Matter in the Universe. See the Potent Method.

2) The study of the anisotropies and of the irregualrities on the Microwave maps and their relation to:

a) The foreground contaminationb) The finger prints of matter on radiation.

3) The epoch of the reionization and the comparison of the quasars observations with numerical simulations and the estimates of the WMAP mission.




The youngest bound objects



The census0.01 0.1 1.00.0030.001

HDM Massive Neutrinos

Photons

Light Neutrinos

Gravity Waves

0.03 0.3

Stars

Gas

Baryons

CDM

Matter

Total


Gamow et al. - Alpher & Herman• A great prevision and no reward ?:

– R.B. Partridge (1995): 3K The Cosmic Microwave Background Radiation.– R.A. Alpher and R. Herman Physics Today August 1988.– Gamow,G. 1956 September Scientific American.– Gamow,G. 1952 The Creation of the Universe, Viking Press– Gamow, G., 1946, Phys. Rev., 70, 572 and Erratum Corrige in 71, 273.

• Alpher and Herman considered the possibility to carry out a radio search to detect the background but were informed by their colleagues observer that the technology at the time was no sensitive enough for that detection.

• Weinberg in Gravitation and Cosmology (1972 Page 510) misses the point when he states :

– “A somewhat more detailed analysis along these lines, carried out in 1950 by Alpher and Herman, gave T0 = 5 degrees. Unfortunately, Alpher and Herman went on to express doubts as to whether this radiation would have survived until the present.”

– When A&H mentioned those doubts, they were discussing cosmic rays and not the Thermal microwave cosmic radiation.


A very Brief Summary - MWB• 1915 General Relativity – The problem.• 1925 The expanding Universe.• 1946 – 1950 Big Bang Nucleosynthesis – Gamow, Alpher & Herman –

Predictions.• 1961 Sandage – Two parameters and a Model. However the evolving

Universe – • 1965 Dicke & Co. – Penzias and Wilson• 1970 Dipole Peebles – Zeldovich• 1970 – 1980 Anisotropies – See the Creta meeting Editors Abell &

Chincarini.• 1990 COBE: FIRAS & COBRA• 1992 COBE DMR Anisotropies

– End of the upper limits era– Low resolution however. Excellent estimate of the Temperature.

• 2000 Boomerang MAXIMA etc Maps and Spherical Harmonics Peak estimate.

• 2002 WMAP High resoluion maps.• ……….. The story continues …


Gamow 1948• If we form the heavy elements from elementary particles, and more precisely

usingProtons and Neutrons

• Then one of the fundamental reaction is that to form Deuterium:n + p => d +

• And this reaction happens at a temperature of about:T ~ 109 degrees

• On a temperature somewhat larger that 109 degress the photons dissociate the Deuterium as soon as it forms.

• That is we must have a T 109. Furthermore we must be able to accumulate Deuterium as a first step to build up heavier elements.

• The density is also critical. I must have a density high enough to allow a reasonable probability for the reaction.

• On the other hand the density must not be too high. I can not overproduce heavy elements since the amount of Hydrogen must remain the highest one as I observe.

• Then I have an interplay between the cosmological model which changes density and Temperature as a function of time and the nuclear reaction rates.


How fast ?

n(t)

v t

Number of encounters = n v t Or\1 enclounter every seconds

I assume 1 encounter=1 reaction

For the reaction to occur at the desired temperature it must be that theThe reaction rate is smaller than the expansion time of the Universe since

otherwise theTemperature decreases and the reaction has not time to occurr.

1/(n v ) < texp or (n v ) >1

We now look at the Cosmology – Note that we are considering theearly phases of the Universe. In this case (t) is very close to 1. That

is I can consider a flat Universe with the term k = 0 :


When

2 22

2

2 4

Tot Rad 2

0r r ,0

2

r rr ,0 0 2

r ,0 0 r

a 8 G kcH 1 ~ 0 ; k and termscana 3 a 3

be disregarded in t he earlyUniverse

8 G ta T; t ta 3 c

By preserving Black Body as we shall show

aT t T

a t

T t T tT a

T a T t

24

r2

62 r

r 2

T8 G3 c

T8 GT t3 c


24

r

4

r 2 2

2 2

4

The solution is

3c32 GT t

t3 T 3t

32 Gt c 32 G

3c 3ct32 G 32 G T

7.56 10-15

For T=109 t = 230.5 sec and trasforming T=109 in velocity:n(t)= 1/( v t ) = 1018 nucleons cm-3

Remember this is strictly valid only for the radiatiodominated Universe. The extension is an approximation


We also have:

12 4

r ,00 r ,0 0 r ,0 02

r 4

12

1 170 0 0

0 0

T t 3ca t a T a T a tT t 32 G3c

32 G

a t 2; t H flat 2.86 10 sec H 72a t 3


The prevision

3 332 2

18 18 50 17

0 0

29 3

a t t 230n n 230 _ s 10 10 2.27 10a t 2.86 10

or

3.8110 g cm

Today we estimate the density of Baryons from nucleosynthesis to be 0.014 < Bh2 <0.026. Assuming 0.02 and Ho=72 we derive a density of:

B = 1.94 10-31 g cm-3


And a Temperature• To obtain a more accurate result we should account for the fact that the

relation between the epansion parameter and the time changes during the later period dominated by radiation. By so doing I would obtain a value rather close [ ~ 5 Degrees] to what is being observed today [2.7 degrees].

• But since we are only making order of magnitude estimates we use the previous relation for the models.

12

0r r ,0 r ,0 r r

0 0

9r ,0 17

a ta tT t T T T t T ta t a t

230T 10 282.86 10



What happens during the expansion• I have a black body radiation B per unit solid angle.• If I divide by h I get the number of photons.• If I divide by c I get the density of radiation and T is the Temperature.• When the volume expand I assume I conserve the number of photons. • This is true and however we should look into the mechanisms capable of

creating and destroying the photons.• What are the mechanisms by which at z < 109 it is possible to create

photons or change their Energy?– Thermal Bremstrahlung.(free – free)

• This is a function of the density of baryons– Compton – Electrons scatter photons – Radiative Compton – A second photon is produced in the e+ scatter.

• These are a function of the Energy and density. The reaction are important only at very high Temperatures and density. Can be disregarded at lower Temperatures.

• I could also use the invariant I/3. Or again by stating that I preserve the law of physics, the BB therefore, and the number of photons, I get again I/3 to be invariant.


3 21 2 1 1

2

3

3

3

2 3

3

3

2 cI B erg s cm ster Hz ; Photons are inVolumeV t

hexp 1kT

2 V tu d4 dN t d

hh c exp 1kT

a t a t a t; d d ; V t V t

a t a t a t

a t a t2 V t

a t a tdN t dN t

a th

a tc exp

kT

2

3

a t 2 V td d

ha t c exp 1kT

1

a tdefining T T

a t


Problem• We observe an object which emits Black Body radiation and is at temperature

T in its reference frame. The object is at redshift z subtending a solid angle d. What is the flux. What is the redshift assuming a Doppler motion rather than a cosmological?

• We will use of the fact that I3 is an invariant.

• I conserve the blackbody spectrum and all I measure is equivalent to a BlackBody spectrum at z=0 with a Temperature of T/(1+z).

• The invariance holds in general, that is it does not make any difference of how we interpret the redshift, we always have the same result.

obs emit 3obs 3 emit

observed obs obs obs emit43 3obs emit

emit 4emit4 4

I IFlux I d d d

1 z

I d T1 z 1 z


Black Body - n/nB

44 15 13 3

34 3r 2

3 42 3

2 x0 0 0

46 4

2 3

photons 30

4u T B T d T 7.56 10 2.7 4.02 10 erg cmc

u T h4.5 10 g cm ; xc kT

2h c 2h kT x dxB T d dh c h e 1exp 1kT

2.77 10 T15

8 c 8 kN dh cexp 1kT

2 2

x0

T kT x dx 398.1h h e 1


0

2 29 229 2 240 0 0

0 ,c 0 H B( 0.02 ) 24

5 2 7 20 0 0 0

B

Matter and Radiation :

3H 1.87 10 h1.87 10 h ; m 1.67 10 ; n8 G 1.67 10

n1.110 h ; 3.6 10 h

n


More about Radiation – Specific Heat

• The energy of each molecule of a monoatomic gas is 3/2 k T.• We indicate with N the number density of molecules. See Landau

and Lifshitz for the definitions.

3V ,m V ,r

V V

5 168V ,m

33 15V ,r

E 3 EC Nk and for a BB C TT 2 T

3 3kN 10 1.38 10C 2 2 1.4 10C T 7.56 10 2.7

This is a result in the same direction of the number of photons versus the number of baryons. To change by 1 degree the Radiation temperature we need about 108 times the energy needed to change by 1 degree the matter.


Entropy

4

3T T 3 3

0 0

3B

3

B

3 23 3

3 x0

Density of Energy u T

du 4 T dT 4Entropy S T a tT T 3

Density of particles n a t

S 4Entropy per baryon T constn 3 n

8 kT x dx Sn T a t constc h e 1 n


The MWB - Observations


The Dipole

• The small value of the anisotropy, =v/c ~ 10-3, simplify the derivation since we can disregard relativistic effects.

• It is not, as often simplified, a simple Doppler effect.• The Doppler effect will increase the energy of the photon

in the direction of the motion of the factor /0 = (1+ cos). On the other hand the interval of frequencies d also increases of the same factor d0 = d0 (1+ cos).

• Since the Temperature is defined in terms of Energy per unit frequency, see for instance the Black Body, the net effect of Doppler is that the Temperature does not change.


The moving observerSee Peebles Physica Review 23 October 1968 Vol 174, Page 2168

• The observer who moves in a certain direction will collect, in the direction of the motion, more photons than the steady observer. The latter colects cdt * Area * density and the former (cdt+v dt cos) *Area * density. That is a factor (1+ cos) difference.

• A second effect acts on the solid angle in the following way and accounting for the effect of aberration, that is by moving the angular position of an object changes.1. I use the relativistic transformation of velocities.2. The observer is moving with a velocity v. The photons collected have

energy E in the range dE3. The solid angle d can be written as d = d sin d = d d(cos). 4. The two observers agree on the Number of photons they collected, that

is dN=dN’.5. The Numbe of photns per unit volume, solid angle and energy interval

is n(E,).6. The observers also agree on the Area A0 of the detector.


Detailsyx

x yx x2 2

x y

x y

x

u /u vu ; u ; the photons come from and .u u1 v 1 vc c

u c Cos ; u c Sin

c Cos v c Sin /u ; u andCos Cos1 v 1 vc c

vCosu CoscCosCosc 1 Cos1 vc

v


Useful relations

2 2

2

2

2

2

2

Cos 11 cos ' 11 Cos 1 Cos 1 Cos

coscos ; cos cos 1 cos cos1 cos 1 cos

d cos 1d 2 d cos 2 d cos 2 d cosd cos 1 Cos

1d d1 Cos


That isFor the Energy transformation see for instance French S.R. Page 210

2

2 22

xx

0 0

In a similar way

d cos 1d d cos d dd cos 1 cos 1 cos

dt dt ; the photon comes toward mep cosE E vp E 1 v E 1 v ; same forE c

dN n E , dE d dt A cos dN n E, dEd dtA v cos

n E ,

2

22

22

22

2

22

d 1n E, dt1 cos 1 cos1 cos

1 cosdn E, dt1 cos1 cos

1n E, n E,1 cos


Using Planck Equation

2 22 22 22 2

22 222 222

2 c 2 cn E , n E, 1 cos 1 cos

h hexp 1 exp 1kT kT

2 1 cos c2 c 2 c1 cos

h hh 1 cosexp 1 exp 1exp 1kT kTkT

h 1 cos 1 coshkT kT

2

1T T

1 1T T 1 cos T T1 cos 1 cos


Approximating

1 12 22 2

2

0 0

0

1T T for semplicity I use1 cos

seriesf 0 1

cos 1f 0 1 1 21 cos1 cos

f 0 cos

T T f 0 f 0 T 1 cos

T T T T T T cosT cosT


More for fun – A satellite

x vt y h emits signals with frequency

h

x1 x2

Ground

Obs

x=0


The satellite Frame• Two Pulses, from x1 and x2

separated by .• Time to reach the observer are

r1/c and r2/c .

The Observer• Pulse separated by = /

since = 1/• The observer measures a time

separation = r2/c-r1/c+

• Due to the large distance of the satellite and the small sepration between x2 and x1 we can write:

r1-r2 ~(x2-x1) cos

=vcos and = r2/c-r1/c+ = -vcos/c +

= (1-v cos/c) or = / [ (1- cos)]


The observed motion – First Question

• After correction for the motion of the Eart, rotation and revolution about the Sun we obtain the heliocentric velocity which is:

370.60.4 km/sL=264.310.17B=48.50.10

• After making correction for the rotation of our galaxy and for the motion of our galaxy respect the Local Group of galaxies (see later) we have:

VLG-MWB=62722km/sL=2763B=303


K Band =13 mm, =22.8 Ghz


Ka Band =9.1 mm, =33 Ghz


Q Band =7.3 mm, =40.7 Ghz


V Band =4.9 mm, =60.8 Ghz


W Band =3.2 mm, =93.5 Ghz


The Cleaned anisotropy map


The first detailed, all-sky picture of the infant universe. The WMAP image reveals 13 billion+ year old temperature fluctuations (shown as color differences) that correspond to the seeds that grew to become the galaxies. Encoded in the patterns are the answers to many age-old questions, such as the age and geometry of the Universe.

The anisotropy – II Question


The second question

• The seeds of in_homogeneities exist in the Universe since the very beginning and during the period matter and radiation were coupled these in_homogeneities were present both in the radiation and in the matter.

• After decoupling matter perturbations grew or dissipated but the print of the Microwave had to be visible and correlate to the distribution of matter we see now.

• That is the anisotropies are of fundamental importance to the understanding of the Universe, its formation and evolution.

• It is also apparent that in order to see the foot print of galaxies or smaller objects we need a very good resolution.


The resolution of the MWB

• I show on the right Figure the comparison between the COBE satellite and the WMAP.

• COBE with a resolution of about 7 degrees could only show the correlation with the very large structures.

• The relation between angular and linear size is about:

(L)=34.4” (h)(l0/1 Mpc)


The Finger prints – Courtesy of

• We believe that large scale structure in the universe grew out of small perturbations in the early universe through gravitational instability. This implies that the photon-baryon fluid moves in a gravitational potential well before last scattering.

• This assumes a Newtonian representation of perturbations. The response of the fluid to the gravitational potential fluctuations allow us to measure the properties of the fluid in an expanding universe known to be filled with dark matter, which allows us to extract basic cosmological parameters, as well as those of the seed perturbations, which can be used to pin down the nature of large scale structure formation in the universe.


Accuracy of Parameters

Each point is the mean of various observations and it is clearly how small are the observational errors.

This Figure is shown simply to show that by analyzing the distribution of the irregularities, after subtraction of foreground disturbing, but extremely interesting objects, the Cosmological parameters can be derived very accurately.



How it works

• By going back in time, from left to the right, objects and structures become fuzzy and while showing structures they move toward recombination where the decoupling between matter and radiation occurred. Here we find the Microwave background radiation and the foot print left from matter to radiation via the strong coupling due to Thomson scattering.


As a function of scale size

• The plot shows that the value of the density parameter increases as a function of the scale length.

• This is equivalent to say that the Mass to Luminosity ratio increases with scale length.

• An analogous result had been derived long ago (early seventies) by H.J. Rood who showed that the Mass to Luminosity ratio of extragalactic system is a function of the Virial mass.

• The reader search in the literature for this resul and discusses it.



III Question

• And we conclude this part with the problem of reionization. According to the new observation of WMAP the epoch of reionization occurred at zr = 20-9

+10.• Recent observtions of quasars at z > 6 showed that

re_ionization is very near z ~ 7 and this was in reasonable agreement with the observations.

• This need to be discussed and set in a common and reasonable framework.

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The Microwave Background - TBC

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